Let k be an algebraically closed field of characteristic zero. An element F
from k(x_1,...,x_n) is called a closed rational function if the subfield k(F)
is algebraically closed in the field k(x_1,...,x_n). We prove that a rational
function F=f/g is closed if f and g are algebraically independent and at least
one of them is irreducible. We also show that the rational function F=f/g is
closed if and only if the pencil af+bg contains only finitely many reducible
hypersurfaces. Some sufficient conditions for a polynomial to be irreducible
are given.Comment: Added references, corrected some typo
